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COMP5331: Knowledge Discovery and Data Mining
Acknowledgement: Slides modified based on the slides provided
by Lawrence Page, Sergey Brin, Rajeev Motwani and Terry
Winograd, Jon M. Kleinberg
2
PageRank & HITS: Bring Order to the Web
Background and Introduction
Approach – PageRank
Approach – Authorities & Hubs
Motivation and Introduction
Why is Page Importance Rating important?
New challenges for information retrieval on the WorldWide Web. Huge number of web pages: 150 million by1998
1000 billion by 2008
Diversity of web pages: different topics, different quality, etc.
Hard to imagine no ranking algorithms in search
engine.
November 9, 2015 Data Mining: Concepts and Techniques 3
Motivation and Introduction
Hard to imagine no ranking algorithms in search
engine.
November 9, 2015 Data Mining: Concepts and Techniques 4
Motivation and Introduction
Modern search engines may return millions of
pages for a single query. This amount is
prohibitive to preview for human users.
Ranking algorithms will process the search results
and only show the most useful information to the
search engine user.
November 9, 2015 Data Mining: Concepts and Techniques 5
Motivation and Introduction
November 9, 2015 Data Mining: Concepts and Techniques 6
PageRank: History
PageRank was developed by Larry Page (hence
the name Page-Rank) and Sergey Brin.
It is first as part of a research project about a
new kind of search engine. That project started
in 1995 and led to a functional prototype in 1998.
Shortly after, Page and Brin founded Google.
November 9, 2015 Data Mining: Concepts and Techniques 7
Link Structure of the Web
150 million web pages 1.7 billion links
November 9, 2015 Data Mining: Concepts and Techniques 8
Backlinks and Forward links:A and B are C’s backlinks
C is A and B’s forward link
Intuitively, a webpage is important if it has a lot of backlinks.
What if a webpage has only one link off www.yahoo.com?
PageRank: A Simplified Version
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u: a web page
Bu: the set of u’s backlinks
Nv: the number of forward links of page v
c: the normalization factor to make
||R||L1 = 1 (||R||L1= |R1 + … + Rn|)
An example of Simplified PageRank
November 9, 2015 Data Mining: Concepts and Techniques 10
PageRank Calculation: first iteration
An example of Simplified PageRank
November 9, 2015 Data Mining: Concepts and Techniques 11
PageRank Calculation: second iteration
An example of Simplified PageRank
November 9, 2015 Data Mining: Concepts and Techniques 12
Convergence after some iterations
A Problem with Simplified PageRank
November 9, 2015 Data Mining: Concepts and Techniques 13
A loop:
During each iteration, the loop accumulates
rank but never distributes rank to other pages!
An example of the Problem
An example of the Problem
An example of the Problem
Random Walks in Graphs The Random Surfer Model
The simplified model: the standing
probability distribution of a random walk on the graph of the web. simply keeps clicking
successive links at random
The Modified Model
The modified model: the “random surfer”
simply keeps clicking successive links at random, but periodically “gets bored” and
jumps to a random page based on the
distribution of E
Modified Version of PageRank
E(u): a distribution of ranks of web pages that “users” jump to
when they “gets bored” after successive links at random.
An example of Modified PageRank
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Dangling Links
Links that point to any page with no outgoing
links
Most are pages that have not been downloaded yet
Affect the model since it is not clear where
their weight should be distributed
Do not affect the ranking of any other page
directly
Can be simply removed before pagerank
calculation and added back afterwards
PageRank Implementation
Convert each URL into a unique integer and store each hyperlink in a database using the integer IDs
to identify pages
Sort the link structure by ID
Remove all the dangling links from the database
Make an initial assignment of ranks and start iteration
Choosing a good initial assignment can speed up the pagerank
Adding the dangling links back.
Convergence Property PR (322 Million Links): 52 iterations
PR (161 Million Links): 45 iterations
Scaling factor is roughly linear in logn
Convergence Property
The Web is an expander-like graph
Theory of random walk: a random walk on a graph is said to be
rapidly-mixing if it quickly converges to a limiting distribution
on the set of nodes in the graph. A random walk is rapidly-
mixing on a graph if and only if the graph is an expander graph.
Expander graph: every subset of nodes S has a neighborhood
(set of vertices accessible via outedges emanating from nodes in
S) that is larger than some factor α times of |S|. A graph has a
good expansion factor if and only if the largest eigenvalue is
sufficiently larger than the second-largest eigenvalue.
PageRank vs. Web Traffic
Some highly accessed web pages have low
page rank possibly because
People do not want to link to these pages from
their own web pages (the example in their paper is pornographic sites…)
Some important backlinks are omitted
use usage data as a start vector for PageRank.
Hypertext-Induced Topic Search(HITS)
To find a small set of most “authoritative’’ pages
relevant to the query.
Authority – Most useful/relevant/helpful results of a query.
“java’’ – java.com
“harvard’’ – harvard.edu
“search engine’’ – powerful search engines.
November 9, 2015 Data Mining: Concepts and Techniques 25
Hypertext-Induced Topic Search(HITS)
Or Authorities & Hubs, developed by Jon
Kleinberg, while visiting IBM Almaden
IBM expanded HITS into Clever.
Authorities – pages that are relevant and are
linked to by many other pages
Hubs – pages that link to many related
authorities
November 9, 2015 Data Mining: Concepts and Techniques 26
Authorities & Hubs
Intuitive Idea to find authoritative results using
link analysis:
Not all hyperlinks related to the conferral of authority.
Find the pattern authoritative pages have:
Authoritative Pages share considerable overlap in
the sets of pages that point to them.
November 9, 2015 Data Mining: Concepts and Techniques 27
Authorities & Hubs
First Step:
Constructing a focused subgraph of the WWW
based on query
Second Step
Iteratively calculate authority weight and hub
weight for each page in the subgraph
November 9, 2015 Data Mining: Concepts and Techniques 28
Constructing a focused subgraph
November 9, 2015 Data Mining: Concepts and Techniques 29
Constructing a focused subgraph
November 9, 2015 Data Mining: Concepts and Techniques 30
Constructing a focused subgraph
November 9, 2015 Data Mining: Concepts and Techniques 31
Constructing a focused subgraph
November 9, 2015 Data Mining: Concepts and Techniques 32
Computing Hubs and Authorities
Rules:
A good hub points to many good authorities.
A good authority is pointed to by many good hubs.
Authorities and hubs have a mutual
reinforcement relationship.
November 9, 2015 Data Mining: Concepts and Techniques 33
Computing Hubs and Authorities
November 9, 2015 Data Mining: Concepts and Techniques 34
Example (no normalization)
November 9, 2015 Data Mining: Concepts and Techniques 35
Example (no normalization)
November 9, 2015 Data Mining: Concepts and Techniques 36
Example (no normalization)
November 9, 2015 Data Mining: Concepts and Techniques 37
Example (no normalization)
November 9, 2015 Data Mining: Concepts and Techniques 38
Example (no normalization)
November 9, 2015 Data Mining: Concepts and Techniques 39
The Iterative Algorithm
November 9, 2015 Data Mining: Concepts and Techniques 40
The Iterative Algorithm
November 9, 2015 Data Mining: Concepts and Techniques 41
The Iterative Algorithm
November 9, 2015 Data Mining: Concepts and Techniques 42
The Iterative Algorithm
November 9, 2015 Data Mining: Concepts and Techniques 43
A Statistical View of HITS
November 9, 2015 Data Mining: Concepts and Techniques 44
A Statistical View of HITS
The weight of authority equals the contribution of
transforming the dataset to first principal
component.
Importance of this vector for the distribution of
whole dataset.
From the statistical view:
HITS can be implemented by PCA
HITS is different from clustering using dimensionality reduction.
The number of samples of PCA is limited.
November 9, 2015 Data Mining: Concepts and Techniques 45
PageRank v.s. HITS
November 9, 2015 Data Mining: Concepts and Techniques 46
47
Summary
PageRank is a global ranking of all web pages based on their locations in the web graph structure
PageRank uses information which is external to the web pages – backlinks
Backlinks from important pages are more significant than backlinks from average pages
The structure of the web graph is very useful for information retrieval tasks.
HITS – Find authoritative pages; Construct subgraph; Mutually reinforcing relationship; Iterative algorithm
